Analytic signals and product theorems for Hilbert transforms

Analytic signals are introduced as certain eigenfunctions of the Hilbert transform operator; that is,z(·)is termed "analytic" if and only ifẑ(t) = -jz(t)for allt, whereẑ(·)is the Hilbert transform ofz(·). Similarly, "dual-analytic" signals are defined as solutions of the homogeneous equationû = ju. Using this characterization of analytic signals (shown to be equivalent to the usual definition due to Ville [1]), simple proofs are obtained for all known product theorems of the formf̂g = fĝ, which are useful in the representation and analysis of modulated waveforms. In addition, parallel theorems for the class of dual-analytic and frequency-translated dual-analytic signals are proven.